Heatmap, Flow Coefficient, Local Entropy, Weighted h-index, Redundancy

Five local measures.

Usage

centrality_heatmap(x, mode = "all", ...)

centrality_flow_coefficient(x, ...)

centrality_local_entropy(x, mode = "all", ...)

centrality_weighted_h_index(x, mode = "all", ...)

centrality_redundancy(x, ...)

Arguments

x: Network input (matrix, igraph, network, cograph_network, tna object).
mode: For directed networks: "all" (default), "out" (distances along out-edges), or "in".
...: Additional arguments passed to centrality.

Value

Named numeric vector, one value per node.

Details

heatmap (Duron 2020): Farness minus the mean farness of the neighbors, \(C(v) = f(v) - \frac{1}{k_v} \sum_{u \in N(v)} f(u)\), with \(f\) the sum of hop distances to reachable nodes. Lower is more central. Isolates score NaN. Reproduces Table 1 of the paper.
flow_coefficient (Honey et al. 2007): Among ordered pairs of distinct neighbors, the fraction joined by a two-step path through the node but not by a direct link, as implemented in the Brain Connectivity Toolbox. On an undirected graph it equals one minus the clustering coefficient; it carries new information only on directed graphs. Nodes with fewer than two neighbors score 0.
local_entropy (Nie et al. 2016): \(-\sum_{j \in N(i)} k_j \ln k_j\), as printed by the sources. Always non-positive and more negative for larger, denser neighborhoods, so lower is more central; isolates score 0, the maximum. The original article is closed access; the formula is that of the Zoo and of Omar and Plapper's 2021 survey, which agree.
weighted_h_index (Gao et al. 2019): h-index of the multiset in which each neighbor \(j\) contributes the topological weight \(k_i k_j\) repeated \(k_j\) times. Edge weights on the input play no role.
redundancy (Burt 1992; Borgatti 1997): Mean degree of the node's neighbors within its ego network, \(2 t_i / k_i\); equal to degree minus effective size. Higher = fewer structural holes. Reproduces Borgatti's worked example.

heatmap, local_entropy and weighted_h_index follow mode; the others ignore direction. Edge weights are ignored.

References

Duron, C. (2020). Heatmap centrality: A new measure to identify super- spreader nodes in scale-free networks. PLOS ONE, 15(7), e0235690.

Honey, C. J., Kotter, R., Breakspear, M., & Sporns, O. (2007). Network structure of cerebral cortex shapes functional connectivity on multiple time scales. PNAS, 104(24), 10240-10245.

Nie, T., Guo, Z., Zhao, K., & Lu, Z.-M. (2016). Using mapping entropy to identify node centrality in complex networks. Physica A, 453, 290-297.

Gao, L., Yu, S., Li, M., Shen, Z., & Gao, Z. (2019). Weighted h-index for identifying influential spreaders. Symmetry, 11(10), 1263.

Borgatti, S. P. (1997). Structural holes: Unpacking Burt's redundancy measures. Connections, 20(1), 35-38.

Examples

star5 <- matrix(0, 5, 5)
star5[1, 2:5] <- 1; star5[2:5, 1] <- 1
rownames(star5) <- colnames(star5) <- LETTERS[1:5]
centrality_heatmap(star5)
#>  A  B  C  D  E 
#> -3  3  3  3  3 
centrality_weighted_h_index(star5)
#> A B C D E 
#> 4 4 4 4 4 
centrality_redundancy(star5)
#> A B C D E 
#> 0 0 0 0 0